The Binary Frobenius Theorem and Prime Geography on S³/2I

We derive an exact identity for the Frobenius multiplicity formula on the Poincaré homology sphere S³/2I. The spectrum of the Laplacian is governed by three integer-valued oscillators — vertex (amplitude 48, period 5), face (amplitude 40, period 3), and edge (amplitude 30, period 2) — corresponding to the Hopf fibre decomposition, plus a linear ramp from the generic (counting) channel. We prove that the Frobenius sum at each residue class modulo 30 takes only the values 0 or 120 = |2I|, establishing a binary spectral sieve. Exactly 15 of 30 residues are killed, including all 10 primes below 30 and all 8 numbers coprime to 30. The spectral gap λ₁ = 168 arises at k = 6 = lcm(2,3), where face and edge fibres first re-synchronise constructively. The last killed eigenvalue is λ = 3480 = 120 × 29 = |2I| × (largest prime below 30). These results are derived entirely from integer arithmetic, with no approximations, decimals, or free parameters.